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Chapter 6: Problem 44

Write each rational expression in lowest terms. $$ \frac{5 s-25}{s^{2}-25} $$

Short Answer

Expert verified
\(\frac{5}{s + 5}\)

Step by step solution

01

Factor the Numerator

Identify the greatest common factor (GCF) of the numerator. In this case, the numerator is \(5s - 25\). The GCF of \(5s\) and \(25\) is \(5\). Factor out \(5\) to get: \[5(s - 5)\]
02

Factor the Denominator

Recognize that the denominator \(s^2 - 25\) is a difference of squares. Recall that a difference of squares can be factored as: \[a^2 - b^2 = (a - b)(a + b)\]Applying this, we get: \(s^2 - 25 = (s - 5)(s + 5)\)
03

Simplify the Expression

Rewrite the rational expression with the factored forms of the numerator and denominator: \[\frac{5(s - 5)}{(s - 5)(s + 5)}\]Next, cancel the common factor \((s - 5)\) from the numerator and the denominator. The simplified form is: \[\frac{5}{s + 5}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Polynomials
Factoring polynomials is an essential skill in simplifying rational expressions. It involves breaking down a polynomial into simpler 'factors' that, when multiplied together, give you the original polynomial. Think of it like finding the ingredients of a recipe. For example, the polynomial expression \(5s - 25\) can be factored by finding a common factor for all terms in the expression.
In this case, both terms (\(5s\) and \(25\)) are multiples of \(5\). So, we factor out the \(5\), which gives us:
\[ 5(s - 5) \] That's all there is to it! The factoring process allows us to simplify and work with the polynomial more easily.
Greatest Common Factor
The greatest common factor (GCF) is the largest number that can divide all terms of a polynomial expression without leaving a remainder. It's like finding the largest group size that can split everything evenly.
For the expression \(5s - 25\), the GCF of the terms \(5s\) and \(25\) is \(5\). Once identified, the GCF can be factored out, simplifying the expression: \[ 5(s - 5) \] By factoring out the GCF, we simplify the terms which makes it easier to combine them in other mathematical operations.
Difference of Squares
A difference of squares is a specific type of polynomial that can be easily factored using a handy rule. The difference of squares formula is:
\[ a^2 - b^2 = (a - b)(a + b) \] An example is the expression \(s^2 - 25\). Notice that \(s^2\) is a perfect square (\(s \cdot s\)), and \(25\) is also a perfect square (\(5 \cdot 5\)). Using our rule, we can factor it as:
\[ s^2 - 25 = (s - 5)(s + 5) \] Understanding how to identify and factor differences of squares is crucial in simplifying rational expressions. By doing so, you can further simplify and work with the entire rational expression more easily.

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Most popular questions from this chapter

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